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physics
oscillations mechanical waves

Questions and Answers of Oscillations Mechanical Waves

A plane, harmonic, acoustical wave that oscillates in air with an amplitude of 10-6 m has an intensity of 10-2 W/m2. What is the frequency of the sound wave?
Water flows at 7 m/s in a pipe of radius 5 cm. A plate having an area equal to the cross-sectional area of the pipe is suddenly inserted to stop the flow. Find the force exerted on the plate. Take
Two wires of different linear mass densities are soldered together end to end and are then stretched under a tension F (the tension is the same in both wires). The wave speed in the second wire is
A column of precision marchers keeps in step by listening to the band positioned at the head of the column. The beat of the music is for 100 paces per minute. A television camera shows that only the
Hovering over the pit of hell, the devil observes that as a student falls past (with terminal velocity), the frequency of his scream decreases from 842 to 820 Hz.(a) Find the speed of descent of the
A bat flying toward an obstacle at 12 m/s emits brief, high-frequency sound pulses at a repetition frequency of 80 Hz. What is the time interval between the echo pulses heard by the bat?
A tuning fork attached to a stretched wire generates transverse waves. The vibration of the fork is perpendicular to the wire. Its frequency is 400 Hz, and the amplitude of its oscillation is 0.50
A very long wire can be vibrated up and down with a mechanical motor to produce waves traveling down the wire. At the far end of the wire, the traveling waves are absorbed by a clever device that
If a loop of chain is spun at high speed, it will roll like a hoop without collapsing. Consider a chain of linear mass density m that is rolling without slipping at a high speed v0.(a) Show that the
A long rope with a mass per unit length of 0.1 kg/m is under a constant tension of 10N. A motor at the point x = 0 drives one end of the rope with harmonic motion at 5 oscillations per second and an
A heavy rope 3 m long is attached to the ceiling and is allowed to hang freely.(a) Show that the speed of transverse waves on the rope is independent of its mass and length but does depend on the
The linear mass density of a nonuniform wire under constant tension decreases gradually along the wire so that an incident wave is transmitted without reflection. The wire is uniform for – ∞ ≤
In this problem you will derive an expression for the potential energy of a segment of a string carrying a traveling wave (Figure). The potential energy of a segment equals the work done by the
(Multiple choice)(1)A traveling wave passes a point of observation. At this point, the time between successive crests is 0.2 s. Which of the following is true? (a) The wavelength is 5
Consider the equation x = ax. If a = 0, then the solution to x = 0 is of course x(t) = C + Dt. Show that in the limit a → 0, equation (3.2) reduces to this form. Note: a
Is the average (over time) tension in the string of a pendulum larger or smaller than mg? By how much? As usual, assume that the angular amplitude A is small.
A mass on the end of a spring (with natural frequency ω) is released from rest at position x0. The experiment is repeated, but now with the system immersed in a fluid that causes the motion to
A particle of mass m is subject to a force F(t) = me−bt. The initial position and speed are both zero. Find x(t). 6
Derive equation (3.31) by guessing a solution of the form x(t) = A cos ωdt+B sin ωdt in equation (3.29).
Three identical springs and two masses, m and 2m, lie between two walls as shown in Fig. Find the normal modes.
Two identical masses m are constrained to move on a horizontal hoop. Two identical springs with spring constant k connect the masses and wrap around the hoop (see Fig.). One mass is subject
(a) Two identical masses m are constrained to move on a horizontal hoop. Two identical springs with spring constant k connect the masses and wrap around the hoop (see Fig.). Find the normal
Two rectangular wave pulses are traveling in opposite directions along a string. At t = 0, the two pulses are as shown in Figure. Sketch the wave functions for t = 1, 2, and 3s
Repeat Problem 3 for the case in which the pulse on the right isinverted.
Two waves traveling on a string in the same direction both have a frequency of 100 Hz, a wavelength of 2 cm, and an amplitude of 0.02 m. What is the amplitude of the resultant wave if the original
Two sound sources oscillate in phase with the same amplitude A. They are separated in space by 1/3 λ. What is the amplitude of the resultant wave from the two sources at a point that is on the line
Two sound sources oscillate in phase with a frequency of 100 Hz. At a point 5.00 m from one source and 5.85 m from the other, the amplitude of the sound from each source separately is A.(a) What is
With a compass, draw circular arcs representing wave crests originating from each of two point sources a distance d = 6 cm apart for ? = 1 cm. Connect the intersections corresponding to points of
Two loudspeakers are separated by a distance of 6 m. A listener sits directly in front of one speaker at a distance of 8 m so that the two speakers and the listener form a right triangle.(a) Find the
Two speakers separated by some distance emit sound waves of the same frequency. At some point P, the intensity due to each speaker separately is I0. The path distance from P to one of the speakers is
Answer the questions of Problem 12 for a point P’ for which the distance to the far speaker is 1λ greater than the distance to the near speaker. Assume that the intensity at point P’ due to each
Two speakers separated by some distance emit sound waves of the same frequency, but the speakers are out of phase by 90o. Let r1 be the distance from some point to speaker 1 and r2 be the distance
Show that if the separation between two sound sources radiating coherently in phase is less than half a wavelength, complete destructive interference will not be observed in any direction.
A transverse wave of frequency 40 Hz propagates down a string. Two points 5 cm apart are out of phase by π/6.(a) What is the wavelength of the wave?(b) At a given point, what is the phase difference
It is thought that the brain determines the direction toward the source of a sound by sensing the phase difference between the sound waves striking the eardrums. A distant source emits sound of
Sound source A is located at x = 0, y = 0, and sound source B is placed at x = 0, y = 2.4 m. The two sources radiate coherently in phase. An observer at x = 40 m, y = 0 notes that as she walks along
Suppose that the observer in Problem 18 finds herself at a point of minimum intensity at x = 40 m, y = 0. What is then the lowest and next higher frequency of the sources consistent with this
Two point sources that are in phase are separated by a distance d. An interference pattern is detected along a line parallel to the line through the sources and a large distance D from the sources,
Two sound sources radiating in phase at a frequency of 480 Hz interfere such that maxima are heard at angles of 0o and 23o from a line perpendicular to that joining the two sources. Find the
Two loudspeakers are driven in phase by an audio amplifier at a frequency of 600 Hz. The speakers are on the y axis, one at y = + 1.00 m and the other at y = -1.00 m. A listener begins at y = 0 a
Two sound sources driven in phase by the same amplifier are 2 m apart on the y?axis. At a point a very large distance from the y?axis, constructive interference is first heard at an angle ?1 = 0.140
Two identical sound sources have a frequency of 500 Hz. The coordinates of the sources are (0, 1 m) and (0, 1 m) A detector 80 m from the origin is free to revolve in the xy plane with a radius
A radio telescope consists of two antennas separated by a distance of 200 m. Both antennas are tuned to a particular frequency, such as 20 MHz. The signals from each antenna are fed into a common
When a violin string is played (without fingering) simultaneously with a tuning fork of frequency 440 Hz, beats are heard at the rate of 3 per second. When the tension in the string is increased
When two tuning forks are struck simultaneously, 4 beats per second are heard. The frequency of one fork is 500 Hz.(a) What are the possible values for the frequency of the other fork?(b) A piece of
A string fixed at both ends is 3 m long. It resonates in its second harmonic at a frequency of 60 Hz. What is the speed of transverse waves on the string?
A string 3 m long and fixed at both ends is vibrating in its third harmonic. The maximum displacement of any point on the string is 4 mm. The speed of transverse waves on this string is 50 m/s.(a)
Calculate the fundamental frequency for a 10-m organ pipe that is(a) Open at both ends, and(b) Closed at one end.
A steel wire having a mass of 5 g and a length of 1.4 m is fixed at both ends and has a tension of 968 N.(a) Find the speed of transverse waves on the wire.(b) Find the wavelength and frequency of
A rope 4 m long is fixed at one end; the other end is attached to a light string so that it is free to move. The speed of waves on the rope is 20 m/s. Find the frequency of(a) The fundamental,(b) The
A piano wire without windings has a fundamental frequency of 200 Hz. When it is wound with wire, its linear mass density is doubled. What is its new fundamental frequency, assuming that the tension
The normal range of hearing is about 20 to 20,000 Hz. What is the greatest length of an organ pipe that would have its fundamental note in this range if(a) It is closed at one end, and(b) It is open
The length of the B string on a certain guitar is 60 cm. Its fundamental is at 247 Hz.(a) What is the speed of transverse waves on the string?(b) If the linear mass density of the guitar string is
The wave function y(x, t) for a certain standing wave on a string fixed at both ends is given by y(x, t) = 4.2 sin 0.20x cos 300t , where y and x are in centimeters and t is in seconds.(a) What are
The wave function y(x, t) for a certain standing wave on a string fixed at both ends is given by y(x, t) = (0.05 m) sin 2.5 m-1 x cos 500 s-1 t.(a) What are the speed and amplitude of the two
A 2.51-m-long string has the wave function given in Problem 46.(a) Sketch the position of the string at the times t = 0, t = ¼ T, t = ½ T, and t = ¾ T, where T = 1/f is the period of the
A string fixed at one end only is vibrating in its fundamental mode. The wave function is y(x, t) = 0.02 x sin 2.36x cos 377t , where x and y are in meters and t is in seconds.(a) What is the
Three successive resonance frequencies for a certain string are 75, 125, and 175 Hz.(a) Find the ratios of each pair of successive resonance frequencies.(b) How can you tell that these frequencies
The space above the water in a tube like that shown in Example 16-8 is 120 cm long. Near the open end, there is a loudspeaker that is driven by an audio oscillator whose frequency can be varied from
A 460-Hz tuning fork causes resonance in the tube in Example 16-8 when the top of the tube is 18.3 and 55.8 cm above the water surface.(a) Find the speed of sound in air.(b) What is the end
At 16oC, the fundamental frequency of an organ pipe is 440.0 Hz. What will be the fundamental frequency of the pipe if the temperature increases to 32oC? Would it be better to construct the pipe with
A violin string of length 40 cm and mass 1.2 g has a frequency of 500 Hz when it is vibrating in its fundamental mode.(a) What is the wavelength of the standing wave on the string?(b) What is the
The G string on a violin is 30 cm long. When played without fingering, it vibrates at a frequency of 196 Hz. The next higher notes on the C-major scale are A (220Hz), B (247 Hz), C (262 Hz), and D
A string with a mass density of 4 x 10-3 kg/m is under a tension of 360 N and is fixed at both ends. One of its resonance frequencies is 375 Hz. The next higher resonance frequency is 450 Hz.(a) What
A string fastened at both ends has successive resonances with wavelengths of 0.54 m for the nth harmonic and 0.48 m for the (n + 1)th harmonic.(a) Which harmonics are these?(b) What is the length of
A rubber band with an unstretched length of 0.80 m and a mass of 6 x 10-3 kg stretches to 1.20 m when under a tension of 7.60 N. What is the fundamental frequency of oscillation of this band when
The strings of a violin are tuned to the tones G, D, A, and E, which are separated by a fifth from one another. That is, f(D) = 1.5f(G), f(A) = 1.5f(D) = 440 Hz, and f(E) = 1.5f(A). The distance
To tune a violin, the violinist first tunes the A string to the correct pitch of 440 Hz and then bows two adjoining strings simultaneously and listens for a beat pattern. While bowing the A and E
(a) For the wave function given in Problem 48, find the velocity of a string segment at some point x as a function of time.(b) Which point has the greatest speed at any time? What is the maximum
A student carries a small oscillator and speaker as she walks very slowly down a long hall. The speaker emits a sound of frequency 680 Hz which is reflected from the walls at each end of the hall.
Assume that the rubber band of Problem 57 behaves like an ideal spring. The band is attached to two posts whose separation D can be varied.(a) Derive an expression for the frequency of the
A 2-m string is fixed at one end and is vibrating in its third harmonic with amplitude 3 cm and frequency 100 Hz.(a) Write the wave function for this vibration.(b) Write an expression for the kinetic
Information for use by computers is transmitted along a cable in the form of short electric pulses at the rate of 107 pulses per second.(a) What is the maximum duration of each pulse if no two pulses
A tuning fork of frequency f0 begins vibrating at time t = 0 and is stopped after a time interval ?t. The waveform of the sound at some later time is shown as a function of x. Let N be the
A musical instrument consists of drinking glasses partially filled with water that are struck with a small mallet. Explain how this works.
During an organ recital, the air compressor that drives the organ pipes suddenly fails. An enterprising physics student in the audience comes to the rescue by connecting a tank of pure nitrogen gas
The following instructions are given for connecting stereo speakers to an amplifier so that they are in phase: “After both speakers are connected, play a monophonic record or program with the bass
Middle C on the equal-temperament scale used by modern instrument makers has a frequency of 261.63 Hz. If a 7-g piano wire that is 80 cm long is to be tuned so that 261.63 is its fundamental
The ear canal, which is about 2.5 cm long, roughly approximates a pipe that is open at one end and closed at the other.(a) What are the resonance frequencies of the ear canal?(b) Describe the
A 4-m-long, 160-g rope is fixed at one end and is tied to a light string at the other end. Its tension is 400 N.(a) What are the wavelengths of the fundamental and the next two harmonics?(b) What are
The shortest pipes used in organs are about 7.5 cm long.(a) What is the fundamental frequency of a pipe this long that is open at both ends?
Two waves from two coherent sources have the same wavelength λ, frequency ώ, and amplitude A. What is the path difference if the resultant wave at some point has amplitude A?
A 35-m string has a linear mass density of 0.0085 kg/m and is under a tension of 18 N. Find the frequencies of the lowest four harmonics if(a) The string is fixed at both ends, and(b) The string is
You find an abandoned mine shaft and wish to measure its depth. Using an audio oscillator of variable frequency, you note that you can produce successive resonances at frequencies of 63.58 and 89.25
A string 5 m long that is fixed at one end only is vibrating in its fifth harmonic with a frequency of 400 Hz. The maximum displacement of any segment of the string is 3 cm.(a) What is the wavelength
The wave function for a standing wave on a string is described by y(x, t) = 0.02 sin 4px cos 60pt, where y and x are in meters and t is in seconds. Determine the maximum displacement and maximum
A 2.5-m-long wire having a mass of 0.10 kg is fixed at both ends and is under tension of 30 N. When the nth harmonic is excited, there is a node 0.50 m from one end.(a) What is n?(b) What are the
In an early method of determining the speed of sound in gases, powder was spread along the bottom of a horizontal, cylindrical glass tube. One end of the tube was closed by a piston that oscillated
In a lecture demonstration of standing waves, a string is attached to a tuning fork that vibrates at 60 Hz and sets up transverse waves of that frequency on the string. The other end of the string
Three successive resonance frequencies in an organ pipe are 1310, 1834, and 2358 Hz.(a) Is the pipe closed at one end or open at both ends?(b) What is the fundamental frequency?(c) What is the length
A wire of mass 1 g and length 50 cm is stretched with a tension of 440 N. It is then placed near the open end of the tube in Example 16-8 and stroked with a violin bow so that it oscillates at its
On a windy day, a drain pipe will sometimes resonate. Estimate the resonance frequency of a drain pipe on a single story house. How much might this frequency change from winter to summer in your
A 50-cm-long wire fixed at both ends vibrates with a fundamental frequency f0 when the tension is 50 N. If the tension is increased to 60 N, the fundamental frequency increases by 5 Hz, and a further
A standing wave on a rope is represented by the following wave function: where x and y are in meters and t is in seconds. (a) Write wave functions for two traveling waves that when superimposed will
Two identical speakers emit sound waves of frequency 680 Hz uniformly in all directions. The total audio output of each speaker is 1 mW. A point P is 2.00 m from one speaker and 3.00 m from the
Three waves with the same frequency, wavelength, and amplitude are traveling in the same direction. The three waves are given by Find the resultantwave.
(a) Show that if the temperature changes by a small amount ∆T, the fundamental frequency of an organ pipe changes by approximately ∆f, where ∆f / f = ½ ∆T / T.(b) Suppose an organ pipe that
Two traveling wave pulses on a string are represented by the wave functions and where x is in meters and t is in seconds. (a) Sketch each wave function separately as a function of x at t = 0, and
The kinetic energy of a segment ?m of a vibrating string is given by (a) Find the total kinetic energy of the nth mode of vibration of a string of length L fixed at both ends. (b) Give the maximum
(a) Show that when the tension in a string fixed at both ends is changed by a small amount dF, the frequency of the fundamental is changed by approximately df, where df/f = ½ dF/F. Does this result

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